A230227 -id:A230227 - cp's OEIS frontend

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 5, 1, 5, 5, 3, 4, 5, 3, 2, 6, 4, 3, 6, 3, 3, 6, 3, 5, 6, 3, 6, 5, 4, 4, 9, 5, 4, 9, 5, 3, 9, 4, 4, 6, 4, 5, 6, 5, 5, 10, 4, 8, 10, 3, 7, 12, 3, 6, 11, 5, 7, 8, 3, 4, 6, 6, 4, 7, 2, 7, 9, 2, 10, 9, 3, 9, 8, 3, 5, 14, 8, 4, 12, 5, 5, 11, 5, 6, 8, 3, 8, 7, 4, 9, 11, 3

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
Conjecture verified for 2*n up to 10^9. - Mauro Fiorentini, Jul 08 2023

			a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.

PQ[n_]:=n>3&&PrimeQ[3n-10]
SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A230241 Number of ways to write n = p + q with p, 3p - 10 and (p-1)q - 1 all prime, where q is a positive integer.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 2, 4, 1, 4, 5, 1, 6, 2, 3, 6, 3, 1, 2, 6, 2, 3, 7, 3, 6, 4, 2, 4, 2, 5, 6, 1, 2, 6, 5, 4, 6, 8, 3, 5, 10, 3, 6, 6, 2, 9, 4, 2, 4, 6, 3, 4, 11, 1, 6, 7, 2, 9, 7, 3, 5, 8, 5, 9, 6, 4, 3, 6, 3, 6, 4, 3, 10, 9, 2, 13, 2, 5, 8, 10, 3, 3, 11, 1, 10, 11, 3, 9, 4, 6, 11

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: a(n) > 0 for all n > 5.
This implies A. Murthy's conjecture mentioned in A109909.
We have verified the conjecture for n up to 10^8.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 29 2023

			a(9) = 1 since 9 = 7 + 2 with 7, 3*7-10 = 11, (7-1)*2-1 = 11 all prime.
a(27) = 1 since 27 = 13 + 14, and the three numbers 13, 3*13-10 = 29, (13-1)*14-1 = 167 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A109909, A227908, A227909, A230227, A230230.

a[n_]:=Sum[If[PrimeQ[3Prime[i]-10]&&PrimeQ[(Prime[i]-1)(n-Prime[i])-1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A274505 Primes p such that 3p-10 and 3p+10 are prime numbers.

Original entry on oeis.org

7, 11, 17, 19, 23, 31, 47, 61, 67, 89, 101, 107, 109, 137, 151, 163, 199, 283, 347, 353, 373, 397, 401, 409, 431, 439, 457, 479, 487, 523, 577, 607, 619, 641, 647, 661, 691, 761, 787, 809, 907, 929, 1033, 1087, 1103, 1153, 1201, 1229, 1307, 1319

Offset: 1

Vincenzo Librandi, Jun 30 2016

Intersection of A023211 and A230227.

			7 is a term because 3*7-10 = 11 and 3*7+10 = 31 are primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A023211, A164568, A230227.

[p: p in PrimesUpTo(1500) |IsPrime(3*p-10) and IsPrime(3*p+10)];

Select[Prime[Range[400]], PrimeQ[3 # - 10] && PrimeQ[3 # + 10] &]

cp's OEIS Frontend

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A230241 Number of ways to write n = p + q with p, 3p - 10 and (p-1)q - 1 all prime, where q is a positive integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A274505 Primes p such that 3p-10 and 3p+10 are prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

cp's OEIS Frontend

A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A230241 Number of ways to write n = p + q with p, 3*p - 10 and (p-1)*q - 1 all prime, where q is a positive integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A274505 Primes p such that 3*p-10 and 3*p+10 are prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A230230 Number of ways to write 2n = p + q with p, q, 3p - 10, 3*q + 10 all prime.

A230241 Number of ways to write n = p + q with p, 3p - 10 and (p-1)q - 1 all prime, where q is a positive integer.

A274505 Primes p such that 3p-10 and 3p+10 are prime numbers.